New Cryptographic Systems Using Pairing with Errors

ABSTRACT

Using the same mathematical principle of paring with errors, which can be viewed as an extension of the idea of the LWE problem, this invention gives constructions of a new key exchanges system, a new key distribution system and a new identity-based encryption system. These new systems are efficient and have very strong security property including provable security and resistance to quantum computer attacks.

BACKGROUND

The present disclosure claims priority to the U.S. provisional patent application with Ser. No. 61/623,272, entitled “New methods for secure communications and secure information systems”, filed Apr. 12, 2012 and PCT application with the same title and the PCT number PCT/CN2013/074053 filed on Apr. 11, 2013, which is incorporated herein by reference in its entirety and for all purposes.

This invention is related to the construction of cryptographic systems, in particular, key exchange (KE) systems, key distribution (KD) systems and identity-based-encryption (IBE) systems, which are based on essentially the same mathematical principle, pairing with errors.

In our modern communication systems like Internet, cell phone, and etc, to protect the secrecy of the information concerned, we need to encrypt the message. There are two different ways to do this. In the first case, we use symmetric cryptosystems to perform this task, where the sender uses the same key to encrypt the message as the key that the receiver uses to decrypt the message. Symmetric systems demand that the sender and the receiver have a way to exchange such a shared key securely. In an open communication channel without any central authority, like wireless communication, this demands a way to perform such a key exchange (KE) in the open between two parties. In a system with a central server, like a cell phone system within one cell company, this demands an efficient and scalable key distribution (KD) system such that any two users can derive a shared key via the key distribution (KD) system established by the central server. Therefore it is important and desirable that we have secure and efficient KE systems and KD systems. The first KE system was proposed by Diffie and Hellman [DiHe], whose security is based on the hardness of discrete logarithm problems. This system can be broken by future quantum computers as showed in the work of Shor [SHO]. There are many key-distribution systems including the system using pairing over quadratic forms [BSHKVY], and the one based on bilinear paring over elliptic curves by Boneh and Boyen (in U.S. Pat. No. 7,590,236). But the existing systems have either the problem of computation efficiency or scalability. For instance, the bilinear paring over elliptic curves is very computationally intensive.

In the second case, we use asymmetric systems, namely public key cryptographic systems, for encryption, where the receiver has a set of a public key and a private key, and the sender has only the public key. The sender uses the public key to encrypt messages, the receiver uses the private key to decrypt the messages and only the entity who has the private key can decrypt the messages. In an usual public key system, we need to make sure the authenticity of the public keys and therefore each public key needs to have a certificate, which is a digital signature provided by a trusted central authority. The certificate is used to verify that the public key belongs to the legitimate user, the receiver of a message. To make public key encryption system fully work, we need to use such a system, which is called a public key infrastructure (PKI) system.

In 1984, Shamir proposed another kind of public key encryption system [SHA]. In this new system, a person or an entity's public key is generated with a public algorithm from the information that can identify the person or the entity uniquely. For example, in the case of a person, the information may include the person's name, residential address, birthday, finger print information, e-mail address, social security number and etc. Since the public key is determined by the public information that can identify the person, this type of public key cryptosystem is called an identity-based encryption (IBE) system.

There are a few Identity-based-encryption (IBE) public key cryptosystems, and currently, the (best) one being practically used is the IBE system based on bilinear paring over elliptic curves invented by Boneh and Franklin (in U.S. Pat. No. 7,113,594). In IBE systems, a sender encrypts a message for a given receiver using the receiver's public key based on the identity of the receiver. The receiver decrypts the message using the the receiver's private key. The receiver obtains the private key from a central server, which has a system to generate and distribute the IBE private key for the legitimate user securely. An IBE system does not demand the sender to search for the receiver's public key, but rather, a sender in an IBE system derives any receiver's corresponding public key using an algorithm on the information that identifies the receiver, for example, an email address, an ID number or other information. Current IBE systems are very complicated and not efficient in terms of computations, since the bilinear paring over elliptic curves is very computationally intensive. These systems based on pairing over elliptic curves can also be broken efficiently if we have a quantum computer as showed in the work of Shor [SHO]. There are also constructions based on lattices, but those are also rather complicated systems for applications [ABB] [ABVVW] [BKPW]. Therefore it is important and desirable that we have secure and efficient IBE systems.

Clearly, there are still needs for more efficient and secure KE, KD and IBE systems for practical applications.

BRIEF SUMMARY OF THE INVENTION

This invention first contains a novel method for two parties A and B to perform an secure KE over an open communication channel. This method is based on the computation of pairing of the same bilinear form in two different ways but each with different small errors. In the KE process, each users will choose a private matrix S_(A), S_(B) respectively with small entries following certain error distributions secretly and a public matrix M randomly. Then each user will compute the multiplication of the user's secret matrix with the publicly chosen matrix but with small errors, exchange the new matrices, and then perform the computation of pairing of S_(A) and S_(B) over the same bilinear form based on M in two different ways but each with different small errors. This kind of mathematical computation is called pairing with errors. The shared key is derived from the pairings with a rounding technique. This method can be viewed as an extension of the idea of the learning with errors (LWE) problem discovered by Regev in 2005 [Reg]. The security of this system depends the hardness of certain lattice problem, which can be mathematically proven hard [DiLi]. This system involves only matrix multiplication and therefore is very efficient. Such a system can also resist the future quantum computer attacks.

This invention second contains a novel method to build a KD system with a central server or authority. In this system, the central server or authority assigns each user i a public ID as a matrix A_(i) with small entries or establish the ID of each user as a matrix A_(i) with small entries following certain error distributions with the information that can identify the user uniquely, and, in a secure way, gives each user a private key based on certain multiplication of this ID matrix with the central server or authority's secret master key M, another matrix, but with small errors. Then any two users in the system will compute the pairing of the two ID matrices of the users with the same bilinear form based on the master key matrix M in two different ways but each with different small errors to derive a shared key between these two users with certain rounding technique. This method can be viewed as an extension of the idea of the learning with error problem discovered by Regev in 2005 [Reg]. The security of this system depends on the hardness of the problem related to pairing with errors. This system involves only matrix multiplication and therefore is very efficient.

This invention third contains a novel method to build a IBE system with a central server or authority. In this system, the central server or authority assigns each user i a public ID A_(i) as a matrix with small entries following certain certain error distributions or establish the ID of each user as a matrix with small entries following certain certain error distributions with the information that can identify the user uniquely. Each user is given by the central server or authority a private key S_(i) based on certain multiplication of this ID matrix with the central server or authority's master private key S, another matrix, but with errors related to one part of the master public key M, another matrix. The central server or authority will establish another half of the mater key as the multiplication of M and S with small errors, which we call M₁. Then any user who wishes to send the user i a message in the system will compute public key of i which consists of M and a paring of M and A_(i) of the bilinear form based on the master secret key matrix S, then encrypt the message using the encryption system based on the MLWE problem, and the user i will use the secret key S_(i) to decrypt the message. This method can be viewed as an extension of the idea of the learning with error problem discovered by REGEV in 2005. The security of this system depends the harness of certain lattice problem, which can be mathematically proven hard. This system involves only matrix multiplication and therefore is very efficient.

In our constructions, we can replace matrices by elements in ideal lattice, and we can also use other type of rounding techniques. We can also build the system in a distributed way where several servers can work together to build KD and IBE systems.

In short, we use the same mathematical principle of paring with errors, which can be viewed as an extension of the idea of the LWE problem, to build secure and more efficient KE, KD and IBE systems.

Though this invention has been described with specific embodiments thereof, it is clear that many variations, alternatives, modifications will become apparent to those who are skilled in the art of cryptography. Therefore, the preferred embodiments of the invention as set forth herein, are intended to be illustrative, not limiting. Various changes may be made without departing from the scope and spirit of the invention as set forth herein and defined in the claims. The claims in this invention are based on the U.S. provisional patent application with Ser. No. 61/623,272, entitled “New methods for secure communications and secure information systems”, filed Apr. 12, 2012, only more technical details are added.

DETAILED DESCRIPTION OF THE INVENTION

1.1 The Basic Idea of Pairing with Errors

The learning with errors (LWE) problem, introduced by Regev in 2005 [Reg], and its extension, the ring learning with errors (RLWE) problem [LPR] have broad application in cryptographic constructions with some good provable secure properties. The main claim is that they are as hard as certain worst-case lattice problems and hence the related cryptographic constructions.

A LWE problem can be described as follows. First, we have a parameter n, a (prime) modulus q, and an error probability distribution n on the finite ring (field) F_(q) with q elements. To simplify the exposition, we will take q to be a odd prime and but we can also work on any whole number except that we may need to make slight modifications.

In F_(q), each element is represented by the set {−(q−1)/2, . . . , 0, . . . , (q−1)/2}. In this exposition, by “ an error” distribution, we mean a distribution we mean a distribution such that there is a high probability we will select an element, which is small. There are many such selections and the selection directly affect the security of the system. One should select good error distribution to make sure the system works well and securely.

Let Π_(S,κ), on F_(q) be the probability distribution obtained by selecting an element A in F_(q) ^(n) randomly and uniformly, choosing e ∈F_(q) according to κ, and outputting (A, <A, S>+e), where + is the addition that is performed in F_(q). An algorithm that solves the LWE problem with modulus q and error distribution κ, if, for any S in F_(q) ^(n), with an arbitrary number of independent samples from Π_(S,κ), it outputs S (with high probability).

To achieve the provable security of the related cryptographic constructions based on the LWE problem, one chooses q to be specific polynomial functions of n, that is q is replaced by a polynomial functions of n, which we will denote as q(n), κ to be certain discrete version of normal distribution centered around 0 with the standard deviation σ=αq≧√{square root over (n)}, and elements of F_(q) are represented by integers in the range [−(q 1)/2, (q 1)/2)], which we denote as κ_(σ).

In the original encryption system based on the LWE problem, one can only encrypt one bit a time, therefore the system is rather inefficient and it has a large key size. To further improve the efficiency of the cryptosystems based on the LWE problem, a new problem, which is a LWE problem based on a quotient ring of the polynomial ring F_(q)[x] [LPR], was proposed. This is called the ring LWE (RLWE) problem. In the cryptosystems based on the RLWE problem, their security is reduced to hard problems on a subclass of lattices, the class of ideal lattices, instead of general lattices.

Later, a new variant of LWE was proposed in [ACPS]. This variant of the LWE problem is based on the LWE problem. We will replace a vector A with a matrix A of size m×n, and S also with a matrix of size n×1, such that they are compatible to perform matrix multiplication A×S. We also replace e with a compatible matrix of size m×1. We will work on the same finite field with q elements.

To simplify the exposition, we will only present, in detail, for the case where A is a square matrices of the size n×n and, S and e of the size n×1.

Let Π_(S,κn), over F_(q) be the probability distribution obtained by selecting an n×n matrix A, whose each entry are chosen in F_(q) uniformly and independently, choosing e as a n×1 vector over F_(q) with entries chosen according to certain error distribution κ_(n), for example, each entries follows an error distribution n independently, and outputting (A, A×S+e), where + is the addition that is performed in F_(q) ^(n). An algorithm that solves a LWE with modulus q and error distribution κ_(n), if, for any vector S in F_(q) ^(n), with any number of independent sample(s) from Π_(S,κ) _(n) , it outputs S (with high probability).

For the case that we choose a small S, namely entries of S are chosen independently according to also the error distribution κ_(n), we call this problem a small LWE problem (SLWE). If we further impose the condition A to be symmetric, we call it a small symmetric LWE problem (SSLWE). If we choose the secret S randomly and independently from the set −z, . . . , 0, 1 . . . , z with z a fixed small positive integer, we call such a problem uniformly small LWE problem (USLWE).

For practical applications, we can choose S and e with different kind of error distributions.

Due to the results in [ACPS], we know If the secret S's coordinates and the error e's entries are sampled independently from the LWE error distribution κ_(σ), the corresponding LWE problem is as hard as LWE with a uniformly random secret S. This shows that the SLWE problem is as hard as the corresponding LWE problem. The same is true for the case of the RLWE problem that if one can solve the Ring LWE problem with a small secret namely the element S being small, then one can solve it with an uniform secret.

We further extend the problem to a full matrix form.

Let Π_(S,κ) _(n) ₂ over F_(q) be the probability distribution obtained by selecting an n×n matrix A, whose each entry are chosen in F_(q) uniformly and independently, choosing e as a n×n matrix over F_(q) with entries following certain error distribution κ_(n) ₂ , for example, an distribution chosen according to the error distribution n independently, and outputting (A, A×S+e), where + is the addition that is performed in F_(q) ^(n) ² . An algorithm that solves a LWE with modulus q and error distribution κ_(n) ₂ , if, for any n×n matrix S in F_(q) ^(n), with any number of independent sample(s) from Π_(S,κ) _(n) ₂, it outputs S (with a high probability).

We call this problem matrix LWE problem(MLWE). For the case where we choose a small S, namely entries of S also follows the error distribution κ_(n) ₂ , we call this problem a small MLWE problem (SMLWE). If we further impose the condition A to be symmetric, we call it a small symmetric MLWE problem (SSMLWE). If we choose the secret S randomly and independently from the set −z, . . . , 0, 1 . . . , z with z a fixed small positive integer, we call such a problem uniformly small MLWE problem (USMLWE). It is clear the MLWE problem is nothing but put n LWE problem together and sharing the same matrices. Therefore it is as hard as the corresponding LWE problem.

We can use different error distributions for S and e.

The mathematical principle behind our construction comes from the fact of associativity of matrices multiplications of three matrices A, B and C:

A×B×C=(A×B)×C=A×(B×C).

Such a product can be mathematically viewed as computing the bilinear paring of the row vectors of A with column vectors of C.

For two matrices A and B with small entries following certain error distributions, for example, with entries following some error distributions, instead of computing this product directly, we can first compute

AB+E_(a),

then compute

(AB+E _(A))C or (AB+E _(A))C+E _(AC),

or we will compute

BC+E_(C),

then compute

A(BC+E _(c)) or (AB+E _(A))C+E _(BC),

where E_(A), E_(B), E_(AC), E_(BC) are matrices with small entries following the same (or different) error distributions. Then we have two way to compute the product ABC with small errors or differences between these two matrices. We call such a computation pairing with errors. All our constructions depends on such a paring with errors and on the fact that the two different paring are close to each other if A and C are also small.

We can mathematically prove the theorem that an MLWE problem is as hard as the corresponding LWE problem with the same parameters. This provides the foundation of the provable security of our constructions

1.2 The Construction of the New KE Systems Based on Paring with Errors

Two parties Alice and Bob decide to do a key exchange (KE) over an open channel. This means that the communication of Alice and Bob are open to anyone including malicious attackers. To simplify the exposition, we will assume in this part all matrices involves are n×n matrices. But they do not have to be like this, and they can be matrices of any sizes except that we need to choose the compatible sizes such that the matrix multiplications performed are well defined.

Their key change protocol will go step by step as follows.

-   -   (1) Alice and Bob will first publicly select F_(q), n and a n×n         matrix M over F_(q) uniformly and randomly, where q is of size         of a polynomial of n, for example q≈n³, and an error         distribution κ_(n) ₂ to be a distribution over n×n matrices over         F_(q), for example, a distribution that each component are         independent and each component follow certain error distribution         like the discrete error distribution κ_(σ) as in the case of         LWE, namely a discrete normal distribution over F_(q) center         around 0 with standard deviation approximately √{square root         over (n)}. All the information above is public. They jointly and         publicly choose a small (prime) integer t (t<<n).     -   (2) Then each party chooses its own secret S_(i) (i=A, B) as a         n×n matrix chosen according to the error distribution κ_(n) ₂ ,         e_(i) also as a n×n matrix following the error distribution. For         Alice, she computes

M _(A) =MS _(A) +te _(A),

-   -   where t is a small integer (t<<n).         -   For Bob, he computes

M _(B) =M ^(t) S _(B) +te _(B).

-   -   (3) Both parties exchange M_(i) in the open communication         channel. This means both M_(i) (i=A, B) are public, but keep         S_(i) and e_(i) (i=A, B), secret.     -   (4) Alice computes:

K _(A) =S ^(t) _(A) ×M _(B) =S ^(t) _(A) M ^(t) S _(B) +tS ^(t) _(A) e _(B).

-   -   -   Bob computes:

K _(B) =M ^(t) _(A) ×S _(B) =S ^(t) _(A) M ^(t) S _(B) +te ^(t) _(A) S _(B).

-   -   (5) Both of them will perform a rounding technique to derive the         shared key as follows:         -   (a) Bob will make a list T₁ of all positions of the entries             of K_(B) such that these entries are in the range of             [−(q−1)/4, (q−1)/4] and a list T₂ of all positions which are             not in the range of [−(q−1)/4, (q−1)/4]. Then Bob will send             to Alice the list T₁.         -   (b) Then each party will compute the residues of these             entries modular t in T₁, and for the entries not in T₁,             which is in T₂, they will add (q−1)12 to each entry and             compute the residue modular q first (into the range of             [−(q−1)/4, (q−1)/4]) then the residue modular t. That gives             a shared key between these two users.

The reason that Alice and Bob can derive from K_(A) and K_(B) a shared secret to be the exchanged key via certain rounding techniques as in the case above is exactly that e_(i) and S_(i) are small, therefore K_(A) and K_(B) are close. We call this system a SMLWE key exchange protocol. We can derive the provable security of this more efficient system [Dili].

In term of both communication and computation efficiency, the new system is very good. The two parties need to exchange n² entries in F_(q), and each perform 2n^(2.8) computations (with Strassen fast matrix multiplication [STR]) to derive n² bits if t=2.

S_(i) and e_(i) can follow different kind of error distributions.

We can prove the theorem that if we choose the same system parameters, namely n and q, the matrix SLWE key exchange protocol is provably secure if the error distribution is properly chosen [DiLi]. The proof relies on the the mathematical hardness of the following pairing with error problem.

Assume that we are given

-   -   (1) an n×n matrix M, a prime integer q, a small positive integer         t, and an error distribution κ_(n) and;

M′ _(A) =MS′ _(A) +te _(A) and M′ _(B) =M ^(t) S′ _(B) +te _(B),   (2)

-   -   where e_(i), a n×1 vector follows the error distribution κ_(n)         and the entries of n×1 vectors also follows the same error         distribution;     -   (3) and the fact that

K′ _(B) =M ^(t) _(A) ×S′ _(B)=(S′ _(A))^(t) M ^(t) S′ _(B) +t<e _(A) , S′ _(B)>

-   -   is in the range of [−(q−1)/4, (q−1)/4] or not;         the problem is to find an algorithm to derive

K′ _(A)=(S′_(A))^(t) ×M _(B)=(S′_(A))^(t) M ^(t) S′ _(B) +t<S′ _(A) , e _(B)>

modular t if K′_(B) is in the range of [−(q−1)/4, (q−1)/4], otherwise K′_(A)+(q−1)/2 first modular q then modular t, with a high probability. We call such a problem a pairing with error problem (PEP).

The proof follows from the fact that the SMLWE problem is as hard as the SLWE problem, since the matrix version can be viewed as just assembling multiple SLWE samples into one matrix SLWE sample.

We note here that we can choose also rectangular matrix for the construction as long as we make sure the sizes are matching in terms of matrix multiplications, but parameters need to be chosen properly to ensure the security.

Similarly we can build a key exchange system based on the ring learning with errors problem (RLWE) [LPR], we will a variant of the RLWE problem described in [LNV].

For the RLWE problem, we consider the rings R=Z[x]/f(x), and R_(q)=R/qR, where f(x) is a degree n polynomial in Z[x], Z is the ring of integers, and q is a prime integer. Here q is an odd (prime) and elements in Z_(q)=F_(q)=Z/q are represented by elements: −(q−1)/2, . . . , −1, 0, 1, . . . , (q−1)/2, which can be viewed as elements in 2 when we talk about norm of an element. Any element in R_(q), is represented by a degree n polynomial, which can also be viewed as a vector with its corresponding coefficients as its entries. For an element

a(x)=a ₀ +a ₁ x+. . . +a _(n-1) x ^(n-1),

we define

∥a∥=max|a _(i)|,

the l_(∞) norm of the vector (a₀, a₁, . . . , a_(n-1)) and we treat this vector as an element in Z^(n) and a_(i) an element in Z. We can also choose q to be even positive number and things need slight modification.

The RLWE_(f,q,X) problem is parameterized by an polynomial f(x) of degree n, a prime number q and an error distribution X over R_(q). It is defined as follows.

Let the secret s be an element in R_(q), a uniformly chosen random ring element. The problem is to find s, given any polynomial number of samples of the pair

(a _(i) , b _(i) =a _(i) ×s+e _(i)),

where a_(i) is uniformly random in R_(q) and e_(i) is selected following certain error distribution X.

The hardness of such a problem is based on the fact that the b_(i) are computationally indistinguishable from uniform in R_(q). One can show [LPR] that solving the RLWE_(f,q,X) problem above is known to give us a quantum algorithm that solves short vector problems on ideal lattices with related parameters. We believe that the latter problem is exponentially hard.

We will here again use the facts in [ACPS], [LPR] that the RLWE_(f,q,X) problem is equivalent to a variant where the secret s is sampled from the error distribution X rather than being uniform in Z_(q) and the error element e_(i) are multiples of some small integer t.

To derive the provable security, we need consider the RLWE problem with specific choices of the parameters.

-   -   We choose f(x) to be the cyclotomic polynomial x^(n)+1 for         n=2^(u), a power of two;     -   The error distribution X is the discrete Gaussian distribution         D_(Z) _(n) _(,σ) for some n>>σ>ω(√{square root over (log n)})>1;     -   q=1 (mod 2n) and q a polynomial of n and q≈n³;     -   t a small prime and t<<n<<q.

We can also use other parameters for practical applications.

There are two key facts in the RLWE_(f,q,X) setting defined above, which are needed for our key exchange system.

-   -   (1) The length of a vector drawn from a discrete Gaussian of         with standard deviation a is bounded by σn, namely,

Pr(∥X∥>σn)≦2^(−n+1),

-   -   for X chosen according to X.     -   (2) The multiplication in the ring R_(q) increases from the         norms of the constituent elements in a reasonable scale, that         is,

∥X×Y(mod f(x))∥≦n∥X∥∥Y∥,

-   -   for X, Y ε R_(q) and the norm is the l_(∞) norm defined above.

With the RLWE_(f,q,X) setting above, we are now ready to have two parties Alice and Bob to do a key exchange over an open channel. It goes step by step as follows.

-   -   (1) Alice and Bob will first publicly select all the parameters         for the RLWE_(f,q,X) including q(≈n³ or similar polynomial         functions of n), n, f(x) and X. In addition, they will select a         random element M over R_(q) uniformly. All the information above         is public.     -   (2) Then each party chooses its own secret s_(i) as an element         in R_(q) according to the error distribution X, and e_(i)         independently also as an element following the error         distribution X, but jointly choose a small prime integer t         (t<<n)         -   For Alice, she computes

M _(A) =Ms _(A) +te _(A),

-   -   where t is a small integer (t<<n).         -   For Bob, he computes

M _(B) =Ms _(B) +te _(B).

-   -   (3) Both parties exchange M_(i). This means both M_(i) are         public, but certainly keep s_(i) and e_(i) secret.     -   (4) Alice computes:

K _(A) =s _(A) ×M _(B) =s _(A) Ms _(B) +te _(B) s _(A.)

-   -   -   Bob computes:

K_(B) =M _(A) ×s _(B) =s _(A) Ms _(B) +te _(A) s _(B).

-   -   (5) Both of them will perform a rounding technique to derive the         shared key as follows:         -   (a) Bob will then make a list of size n, and this list             consists of pairs in the form of (i, j), where i=0, . . . ,             n−1, and j=1 if the x^(i) coefficient of K_(B) is in the             range of [−(q−1)/4, (q−1)/4], otherwise j=0.         -   (b) Then Bob will send this list to Alice. Then each will             compute the residue of the corresponding entries modular t             in the following way:             -   for an element of the list (i, j),             -   1) if j=1, each will compute the i-th entry of K_(A) and                 K_(B) modular t respectively;             -   2) if j=0, each will add (q−1)/2 to the i-th entry of                 K_(A) and K_(B) modular q back to range of [−(q−1)/4,                 (q−1)/4], then compute the residues modular t.

We can use different distributions for s_(i) and e_(i).

That will give a shared key between these two users. We call this system a RLWE key exchange system. We can deduce that there is a very low probability of failure of this key exchange system. We note here that the commutativity and the associativity of the ring R_(q) play a key role in this construction.

In terms of security analysis, we can show the provable security of the system following the hardness of the RLWE_(f,q,X) problem by using a similar PEP over the ring R_(q) [DiLi].

Assume that we are given

-   -   a random element M in R_(q), prime integers t, q and the error         distribution X with parameters selected as in the RLWE_(f,q,X)         above;     -   M_(A)=Ms_(A)+te_(A) and M_(B)=Ms_(B)+te_(B), where e_(i) follows         the error distribution X and s_(i) also follows the error         distribution X;     -   and the fact that (K_(B))_(i), the coefficients x^(i) of         K_(B)=M_(A)×s_(B)=s_(A)Ms_(B)+te_(A)s_(B) is in the range of         [−(q−1)/4, (q−1)/4] or not;         the problem is to find an algorithm to derive K_(B) (or K_(A))         modular t or K_(B)+(q−1)/2 (or K_(A)+(q−1)/2) modular q (into         the range of [−(q−1)/4, (q−1)/4]) and then modular t with a high         probability. We call such a problem a pairing with error problem         over a ring(RPE).

It is nearly a parallel extension of the proof of the provable security of the case of SLWE key exchange system to the RLWE key exchange system. We conclude that the RLWE key exchange system is provable secure based on the hardness of the RLWE_(f,q,X) problem.

With the same parameters q and n, this system can be very efficient due to the possibility doing fast multiplication over the ring R_(q) using FFT type of algorithms.

1.3 The Construction of the New KD Systems Based on Paring with Errors

Over a large network, key distribution among the legitimate users is a critical problem. Often, in the key distribution systems, a difficult problem is how to construct a system, which is truly efficient and scalable. For example, in the case of the constructions of [BSHKVY], the system can be essentially understood as that the master key of a central server is a symmetric matrix M of size n×n and each user's identity can be seen as a row vector H_(i) of size n. The central server gives each user the secret H_(i)×M. Then two users can derive the shared key as H_(i)×M×H_(j) ^(t). The symmetric property of M ensures that

H _(i) ×M×H _(j) ^(t) =H _(j) ^(t) ×M×H _(i).

However, large number of users can collaborate to derive the master key. If one can collect enough (essentially n) H_(i)×M, which then can be used to find the master key M and therefore break the system.

We will build a truly scalable key distribution system using the pairing with error with a trusted central server, which can be viewed as a combination of the idea above and the idea of the LWE.

We work again over the finite field F_(q), whose elements are represented by −(q−1)/2, . . . , 0, . . . , (q−1)/2. We choose q≈n³ or other similar polynomial function of n, we choose again κ_(n) ₂ to be an error distribution over the space of n×n matrices, for example, an distribution each component are independent, and each component follows error distribution κ_(σ), the discrete distribution as in the case of LWE, namely a discrete normal distribution over F_(q) centered around 0 with standard deviation approximately √{square root over (n)}. The choice of these parameters can be modified.

The key distribution system is set up step by step as follows.

-   -   (1) We have a central server, which will select a symmetric         randomly chosen n×n matrix S, as a master key, whose entries are         in F_(q):

S=S^(t).

-   -   (2) For each user index as i, the central server gives it a (in         general not symmetric) matrix A_(t) (as an ID) with small         entries following error distribution κ_(n) ₂ . The ID matrix of         each user is public and it can also be generated with         information that can identify the user like email address, name         and etc.     -   (3) For each user, the central server distribute securely a         secret:

E _(i) =A _(i) S+te _(i),

-   -   where e_(i) is a matrix (not symmetric) selected following         certain error distribution, such as κ_(n)2. This is kept private         for each user.

To obtain a secret key shared between the user i and the user j, the user i computes

K _(i) =E _(i) ×A _(j) ^(t) =A _(i) SA _(j) ^(t) +te _(i) A _(j) ^(t);

and the user j computes

K _(j) =A _(i)×(E _(j))^(t) =A _(i) S ^(t) A _(j) ^(t) +tA _(i) e _(j) ^(t) =A _(i) SA _(j) ^(t) +tA _(i) e _(j) ^(t).

This is possible because the IDs are public. They then can use the following simple rounding method to derive a shared key between the two users.

-   -   When the user j wants to establish a shared key with the user i,         the user j will collect all the entries (including their         positions in the matrix) in K_(j) that are in the range of         (−(q−1)/4, (q−1)/4), namely those entries which are closer to 0         than (q−1)/2. Then user j will send to the user i a list of the         positions of the entries in the matrix (only the position not         the values of the entries themselves) that are randomly selected         from the collection, which is tagged by 0, and a list of entries         not in the list tagged by 0. Then the user i will select the         same entries in its own matrix E_(i)×A_(j). Now they have a         shared list of common entry positions, therefore the         corresponding entries of the matrix. Then each user will compute         the residue of these entries modular t tagged by 1 and compute         the residue of the sum of each of these entries tagged by 0 with         (q−1)/2 to build a new identical ordered list of values, which         will be their shared secret key.

Because S symmetric, we have that

A_(i)SA_(j) ^(t)=A_(i)S^(t)A_(j) ^(t),

therefore the user j derives

A_(i)SA_(j) ^(t)+tA_(i)A_(i)e_(j) ^(t).

The difference between the results computed by the two users is:

$\begin{matrix} {{{E_{i} \times A_{j}^{t}} - {A_{i} \times E_{j}^{t}}} = {{A_{i}{SA}_{j}^{t}} + {{te}_{i}A_{j}^{t}} - \left( {{A_{i}{SA}_{j}^{t}} + {{tA}_{i}e_{j}^{t}}} \right)}} \\ {= {{{te}_{i}A_{j}^{t}} - {{tA}_{i}{e_{j}^{t}.}}}} \end{matrix}$

This difference is small since t is small and e_(i)A_(j) ^(t) and A_(i)e_(j) ^(t) are small, which is due to the fact that e_(i), e_(j), A_(i) and A_(j) are all small. This allows us to get a common key for i and j by certain rounding techniques and therefore build a key distribution system.

Since the error terms for both matrices, te_(i)A_(j) and te_(j) ^(t)A_(i), are small, the corresponding selected entries with tag 1 in A_(i)SA_(j) (without the error terms) are essentially within the range of [(−(q−1)/4, (q−1)/4] or very close. Therefore the error terms will not push those selected terms in A_(i)SA_(j) over either (−(q−1)/2 or (q−1)/2), that is when added the error terms, those selected entries will not need any further modular q operation but just add them as integers, since each element is represented as an integer in the range of [(−(q−1)/2(q−1)/2)]. The same argument goes with entries tagged by 0. These ensures that the process give a shared key between these two users.

From the way matrices K_(i), K_(j) are constructed, we know that each entry of K_(i) and K_(j) follows uniform distribution. Therefore we expect that each time the size of the first list selected by the user j from the matrix K_(j) should be around n². Therefore this system can provide the shared secret with enough bits if we choose proper n.

Also we can build a version of this system with none symmetric matrices, in this case, the central serve needs to compute more matrices like A_(i)S+e and A_(i) ^(t)S+e′. Then it is possible, we can do the same kind of key distribution. This system again is less efficient.

On the other hand, since the RLWE problem can be viewed as a specialized commutative version of matrix-based LWE since an element in the ring can be view as a homomorphism on the ring. We can use the RLWE to build a key distribution in the same way.

Now let us look at why this key distribution is scalable. Clearly each user will have a pair A, and E_(i)=A_(i)S+te_(i), and many users together can get many pairs, then to find the secret master key S is to solve the corresponding MLWE problem, except that, in this case, we impose the symmetric condition on the secret S. It is not difficult to argue again that this problem is as hard as a LWE problem, since given a LWE problem, we can convert it also into such a MLWE problem with symmetric secret matrix. Therefore, it is easy to see that this system is indeed scalable.

In terms of the provable security of the system, the situation is similar to the work done in the paper [DiLi]. We can give a provable security argument along the same line.

As we said before, since RLWE can be viewed as a special case MLWE, we will use the RLWE to build a very simple key distribution system.

We will choose the ring R_(q) to be F_(q)[x]/x^(n)+1. To ensure the provable security, we need to choose parameter properly n, q, properly, for exmaple n=2^(k), q=1mod(2n)[LPR]. For provable secure systems, we assume that we will follow the conventional assumptions on these parameters, and the assumption on the error distribution like X in [LPR].

This construction is essentially based on the systems of above. We assume that we have a ring R_(q) with a properly defined learning with error problem on the ring R_(q) with erro distribution X. The problem is defined as follows:

We are given a pair (A, E), where

E=A×S+te′,

A, S where e′ are elements in R, t is small integer, e′ is an error element following the distribution of X, S is a fixed element and A is select randomly following uniform distribution, and the problem is to find the secret S.

With a central server, we can build a simple key distribution system as follows.

-   -   (1) The central server will also select a random element M in         R_(q) following uniform distribution.     -   (2) For each user, the central server will assign an public ID         as A_(i), where A_(i) should be in the form of a chosen small         element in R_(q), namely following an error distribution like X.     -   (3) Each member is given a secret key by the central server:

S _(i) =MA _(S) +te _(e),

-   -   where e_(i) follows an error distribution X.     -   (4) If two user i and j wants to build a shared key, one user,         say i can use the ID matrix of j, namely A_(i), the its secret         key to build a shared key with j by computing

K _(i) =A _(j) ×S _(i) =A _(j) MA _(i) +tA _(j) e _(i),

-   -   and j can use its secret key to build a shared key with i by         computing

K _(j) =A _(i) ×S _(j) =A _(j) MA _(i) +tA _(i) e _(j),

-   -   then derive the shared key with the rounding technique as         follows:         -   (a) i will then make a list of size n, and this list             consists of pairs in the form of (a, b), where a=0, . . . ,             n−1, and b=1 if the x^(a) coefficient of K_(i) is in the             range of [−(q−1)/4, (q−1)/4], otherwise b=0.         -   (b) i will send this list to j. Then each will compute the             residue of the corresponding entries modular t in the             following way: for an element of the list (a, b),             -   1) if b=1, each will compute the a-th entry of K_(i) and                 K_(j) modular t respectively;             -   2) if b=0, each will add (q−1)/2 to the a-th entry of                 K_(i) and K_(j) modular q back to range of [−(q−1)/4,                 (q−1)/4], then compute the residues modular t.

Since A_(i) and e_(i) are small elements in R_(q), we have A_(i)×e_(i) is also small. This ensures that we indeed have a shared secret key. This, therefore, gives an key-distribution system.

Here we use very much the fact that in a RLWE problem that the multiplication is commutative. The key feature of our construction is that it is simple and straight forward. The provable security of the system is also straightforward.

1.4 The Construction of the New IBE Systems Based on Paring with Errors

We will first build a new public key encryption based on MLWE. To build an encryption system, we choose similar parameter q n³ or n⁴ or similar polynomial functions of n, we choose again κ_(n) ₂ to be an error distribution, for example the error distribution with each component are independent, and each component follow the same discrete distribution κ_(σ) as in the case of LWE, namely a discrete normal distribution over F_(q) center around 0 with standard deviation approximately √{square root over (n)}. Surely we can also select high dimensional Gaussian distribution, which should be very convenient for the purpose to provable security. We select this simple distribution to simplify the argument concerning the validity of the encryption system. We can surelt choose other parameters.

With such a setting, we can build an encryption system as in the case of the MLWE problem as follows:

-   -   (1) We select an n×n matrix S, whose entries are small following         an error distribution κ_(n) ₂ , for example, each entries         independently and randomly follows the distribution κ_(σ).     -   (2) In the setting of the MLWE, we will derive one output pair         (A, E), where

E=A×S+e,

-   -   or

E=A×S+te,

-   -   and t is small, t<<n, and they form the public key of our         encryption system. Here e follow certain error distributions,         for example the distribution we use above.     -   (3) S is the private key of the cryptosystem.     -   (4) A message in is represented as n×n matrix with binary         entries of 0, 1 or n×n matrix with entries in the range modular         t, namely 0, 1 . . . , t−1.     -   (5) A sender chooses a n×n small matrix B similar to S namely         following an error distribution κ_(n) ₂ , for example, each         entries independently and randomly follows the distribution         κ_(σ). Then the sender compute the encrypted message as:

(D ₁ , D ₂)=(B×A+e ₁ , B×E+e ₂ +m(q/2)),

-   -   or

((D ₁ , D ₂)=(B×A+te ₁ , B×E+te ₂ +m,

-   -   where e₁ and e₂ are error matrices selected independently         following some error distribution like e.     -   (6) To decrypt, the legitimate, in the first case, computes

D ₂ −D ₁ ×S=(BE+e ₂ +m(q/2)−(BA+e ₁)S)=eE+e ₂ −e ₁ S+m(q/2),

-   -   where everything is done in F_(q), and we can check on each         entry of the matrix, if it is near 0, we output 0, and if it is         near (q−1)/2, we output 1, or we divide them by (q−1)/2         performed as a real number division and round them to 0 or 1 and         the output will be the plaintext in; or in the second case, the         legitimate user computes

D ₂ −D ₁ ×S=(BE+te ₂ +m−(BA+te ₁)S)=,teE+te ₂ −te ₁ S+m,

-   -   then modular t. This will be the plaintext Tn.

A, B, e_(i) can follow different error distributions.

With large n, the output can give us the right plaintext with as high probability as demanded. The reason we could decrypt with high probability comes from the following.

$\begin{matrix} {{D_{2} - {D_{1} \times S}} = {{BE} + e_{2} + {m\left( {q/2} \right)} - {\left( {{BA} + e} \right)S}}} \\ {= {{B \times \left( {{A \times S} + e} \right)} + e_{2} + {m\left( {q/2} \right)} - {\left( {{BA} + e_{1}} \right) \times S}}} \\ {= {{B \times e} + e_{2} - {e_{1} \times S} + {m\left( {q/2} \right)}}} \end{matrix}$

B×e+e₂−e₁×S can be viewed as a error terms, which is determined by the distribution of the following random variable. With proper choice of parameters, like in the case of KE or KD systems, the decryption process will surely return the right answer when n is large enough. The same argument goes with the second case.

One key point of this new method is that on average, we can do the encryption much faster in terms of per bit speed because we can use fast matrix multiplication [CW] to speed up the computation process.

We note here that since matrix multiplication is not commutative, when we multiply two elements, the order is very important, unlike the case of the RLWE related systems.

We can also use the same idea in the ring LWE (RLWE)[LPR] to do encryption, where all the elements are in the ring R_(q), and we have

E=A×S+te,

t is small positive integer and the entries of S is also small following error distribution κ_(n) ₂ . We encrypt a message as

(D ₁ , D ₂)=(BA+te ₁ , BE+te ₂ +m).

Then we decrypt by computing

(BE+te ₂ +m−B(AS+te ₁))(mod t).

This works because

$\begin{matrix} {{D_{2} - {D_{1} \times S}} = {{BE} + {te}_{2} + m - {\left( {{BA} + {t_{1}e_{1}}} \right)S}}} \\ {= {{B \times \left( {{A \times S} + {te}} \right)} + {te}_{2} + m - {\left( {{BA} + {te}_{1}} \right) \times S}}} \\ {= {{{tB} \times e} + {te}_{2} - {{te}_{1} \times S} + m}} \end{matrix}$

Since the error terms are small, by modular t, we certainly should get back the original plaintext.

For the MLWE problem, we surely need to choose the distribution accordingly when we need to obtain the provable security of the system.

There are several versions of identity-based encryption systems based on lattice related problems including the LWE problem [ABB],[ABVVW],[BKPW]. But they all look rather complicated. We can use the MLWE to build an identity-based encryption system.

With a central server, we can build a simple identity-based encryption system as follows.

-   -   (1) The central server will first select a secret n×n matrix S         as the secret master key, where S is selected as a small element         following certain error distribution n2 like error distributions         like in KE and KD systems.     -   (2) The central server will also select a random element M         following uniform distribution or similar distribution, but make         sure that M has an inverse. If we could not find one first time,         we will try again till we find one. We have a high probability         of success to find such a M when q is large. Then the central         serve will compute

M ₁ =MS+te,

-   -   where e is small following certain error distribution κ_(n) ₂ .     -   (3) Then the central server will publicize M and M₁ as the         master public key.     -   (4) For each user, the central server will assign an public ID         as A_(i), where A_(i) is small following certain error         distribution κ_(n) ₂ , and it can be generated from information         that can identify the user.     -   (5) Each member is given a secret key:

S _(i) =SA _(i) +tM ⁻¹ e _(i),

-   -   where e_(i)'s entries are small following the error         distribution n. Surely this is the same as given

MS _(i) =MSA _(i) +te _(i),

-   -   since M is public.     -   (6) Anyone can use the ID, namely A_(i), and the master public         key to build a new public key for the user with ID A_(i), which         is given as the pair (A_(i), B_(i)), where

A_(i)=M

-   -   and

B _(i) =M ₁ A _(i) =MSA _(i) +teA _(i),

-   -   and it is used as the public key to encrypt any message use the         MLWE encryption system above.         This gives an identity based encryption system.

S, A_(i), e_(i), e can also follow different error distributions.

Since A_(i) and e are small, we have A_(i)×e is also small. W also have that

$\begin{matrix} \begin{matrix} {{{MS}_{i} - B_{i}} = {{MS}_{i} - B_{i}}} \\ {= {{M\left( {{SA}_{i} + {{tM}^{- 1}e_{i}}} \right)} - {MSA}_{i} + {teA}_{i}}} \\ {\left. {= {{MSA}_{i} + {{tMM}^{- 1}e_{i}}}} \right) - {MSA}_{i} + {teA}_{i}} \\ {{= {{te}_{i} - {teA}_{i}}},} \end{matrix} & \; \end{matrix}$

Since e, A_(i) and e_(i) are small, e−A_(i)e_(i) is also small and te_(i)−tA_(i)e_(i) is also small. Therefore S_(i) is a solution to a MLWE problem with the pair (A_(i), B_(i)) as the problem input. Therefore S_(i) is indeed a secret key that could be used for decryption. Therefore the construction works. We need to choose parameters properly to ensure security.

The key feature of our construction is that it is simple and straight forward. The provable security of the system is also straightforward.

we can extend this construction using the RLWE problem. We will choose the ring R to be F_(q)[x]/x^(n)+1. To ensure the provable security, we need to choose parameter properly n, q, properly, namely n=2^(k), q=1mod(2n)[LPR]. But we can select other parameters for secure applications.

This construction is directly based on the encryption systems of the RLWE[LPR], namely, we assume that we have a ring R with a properly defined learning with error problem on the ring R. The problem is defined as follows: we are given a pair (A, E), where

E=A×S+te′,

A, S where e′ are elements in R_(q), t is small integer, e′ is an error element following an error distribution X, S is a fixed element and A is select randomly following uniform distribution, and the problem is to find the secret S. We also know that one can build a public key encryption systems using the RLWE problem[LPR], where A, and E serve as the public key, and the secret S, which needs to be small, serves as the private key. We can use the fact that in a ring-LWE problem that the multiplication is commutative.

With a central server, we can build a simple identity-based encryption system as follows.

-   -   (1) The central server will first select a secret S in R as the         secret master key, where S is a selected small element follow         certain error distributions X.     -   (2) The central server will also select a random element M in R         following uniform distribution and make sure that M has an         inverse. If we could not find one first time, we will try again         till we find one. We have a high probability of success to find         such a M when q is large. Then the central serve will computer

M ₁ −MS−te,

-   -   where e is small and follows error distribution X.     -   (3) Then the central server will publicize M and M₁ as the         master public key.     -   (4) For each user, the central server will assign an public ID         as A_(i), where A, is a small element in R_(q), and it follows         error distribution X.     -   (5) Each member is given a secret key:

S _(i) =SA _(i) +tM ⁻¹ e _(i),

-   -   where e_(i) small element in R, and it follow certain error         distribution X. Surely this is the same as given

MS _(i) =MSA _(i) +te _(i),

-   -   since M is public. (6) Anyone can use the ID, namely A_(i), and         the master public key to build a new public key for the user         with ID A_(i), which is given as the pair (A_(i), B_(i)), where

A_(i)=M

-   -   and

B _(i) =A _(i) M ₁ =A _(i) MS+tA _(i) e=MSA _(i) +tA _(i) e,

-   -   and it is used as the public key to encrypt any message.         This gives an identity based encryption system.

The small elements like S, A_(i), e, e_(i) can follow different error distributions.

Since A_(i) and e are small elements in R, we have A_(i)×e is also small. We have that

$\begin{matrix} \begin{matrix} {{{S_{i}A_{i}} - B_{i}} = {{S_{i}M} - B_{i}}} \\ {= {{M\left( {{SA}_{i} + {{tM}^{- 1}e_{i}}} \right)} - {MSA}_{i} + {A_{i}{te}}}} \\ {\left. {= {{MSA}_{i} + {{tMM}^{- 1}e_{i}}}} \right) - {MSA}_{i} + {A_{i}{te}}} \\ {{= {{te} - {{tA}_{i}e_{i}}}},} \end{matrix} & \; \end{matrix}$

which is due to the fact that this is a commutative ring. Since e, A_(i) and e_(i) are small, e−A_(i)e_(i) is also small and te−tA_(i)e_(i) is also small. Therefore S_(i) is a solution to a ring LWE problem with the pair (A_(i), B_(i)) as the problem input. Therefore S_(i) is indeed a secret key that could be used for decryption.

We can build easily a hierarchical IBE system using similar procedure, where each user can server as a central server.

The key feature of our construction is that it is simple, straight forward and efficient. The provable security of the system is also straightforward.

In the all the systems above using pairing with errors over the ring, one may use polynomials in the form of

f(x)=Πf _(i)(x)+g(x),

where each f_(i), g(x) is a extremely sparse matrix with very few terms, for example, 2 or 3 terms none-zero. Using this kind of polynomial can speed up the encryption and decryption computations.

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[STR] V. Strassen, Gaussian Elimination is not Optimal, Numer. Math. 13, p. 354-356, 1969 

1. Method for establishing a key exchange over an open channel between a first party A and a second party B, comprising: (1) openly selecting, by Party A and Party B together, parameters, n, q and small whole number t, (t<<n), where q is an odd prime, and an error distribution n2 to be a distribution over n×n matrix over F_(q), a n×n matrix M over F_(q) uniformly and randomly, where q is of size of a polynomial of n like n³, and elements of F_(q) are represented by integers in the range [−(q−1)/2, (q−1)/2)]; (2) choosing, by each party privately, its own secret matrix S_(i) (i=A, B) a n×n matrix chosen according to the error distribution κ_(n) ₂ , and error matrix e_(i), (i=A, B) as a n×n matrix following the error distribution κ_(n) ₂ ; computing by Party A M _(A) MS _(A) +te _(A), where t is a small integer (t<<n); computing by Party B M _(B) =M ^(t) S _(B) +te _(B). (3) Both parties exchange M_(i) in the open communication channel; (4) computing by Patty A: K _(A) =S ^(t) _(A) ×M _(B) =S ^(t) _(A) M ^(t) S _(B) +tS ^(t) _(A) e _(B); computing by Party B: K_(B) =M ^(t) _(A) ×S _(B) =S ^(t) _(A) M ^(t) S _(B) +te ^(t) _(A) S _(B); (5) performing by both Party A and Party B a rounding technique to derive the shared key, comprising: (a) making by Party B a list T₁ of all positions of the entries of K_(B) such that these entries are in the range of [−(q−1)/4, (q−1)/4] and a list T₂ of all positions which are not in the range of [−(q−1)/4, (q−1)/4], then sending by Party B to Party A the list T₁. (b) computing by each party privately the residues of these entries modular t in T₁, and for the entries not in T₁, which is in T₂, adding (q−1)/2 to each entry and computing the residue modular q first (into the range of [−(q−1)/4, (q−1)/4]) then the residue modular t, which gives a shared key between these two parties.
 2. Method, for a central server, building a key distribution (KD) system, comprising: (1) selecting, by the central server, parameters select parameters, n, q and small whole number t, (t<<n), where q is an odd prime, q is of size of a polynomial of n like n³ and elements of F_(q) are represented by integers in the range [−(q−1)/2, (q−1)/2)], an error distribution κ_(n) ₂ a distribution over n×n matrix over F_(q); and selecting by the central server a symmetric randomly chosen n×n matrix S over F_(q) as a master key, (2) giving, by the central server, to each user index as i, a general matrix A_(i) as an ID with small entries following error distribution κ_(n) ₂ , where the ID matrix of each user is public and the central server have also a choice to generate the ID with information that can identify the user; (3) distributing, by the central server, for each user securely a secret: E _(i) =A _(i) S+te _(i), where e_(i) is a matrix selected following error distribution κ_(n) ₂ and this is kept private for each user; obtaining a secret key shared between the User i and the User j comprising: computing by the User i: K _(i) =E _(i) ×A _(j) ^(t) =A _(i) SA _(j) ^(t) +te _(i) A _(j) ^(t); and computing by the User j K _(j) =A _(i)×(E _(j))^(t) =A _(i) S ^(t) A _(j) ^(t) +tA _(i) e _(j) ^(t) =A _(i) SA _(j) ^(t) +tA _(i) e _(j) ^(t); then two users deriving a shared key between the two users using the following simple rounding method, comprising: assuming the User j wants to establish a shared key with the user i, collecting by the user j all the entries (including their positions in the matrix) in K₃ that are in the range of (−(q−1)/4, (q−1)/4), namely those entries which are closer to 0 than (q−1)/2; sending by the User j to the user i a list of the positions of the entries in the matrix (only the position not the values of the entries themselves) that are randomly selected from the collection, which is tagged by 0, and a list of entries not in the list tagged by 0; then selecting by the user i the same entries in its own matrix E_(i)×A_(j), which gives them a shared list of common entry positions, therefore the corresponding entries of the matrix; then computing by each user the residue of these entries modular t tagged by 1 and compute the residue of the sum of each of these entries tagged by 0 with (q−1)/2, which build a new identical ordered list of values, their shared secret key.
 3. Method, for a central, building an identity-based encryption system, comprising: (1) selecting by the central server parameters, n, q and small whole number t, (t<<n), where q is an odd prime, q is of size of a polynomial of n like n³ and elements of F_(q) are represented by integers in the range [−(q−1)/2, (q−1)/2)], and an error distribution κ_(n) ₂ to be a distribution over n×n matrix over F_(q); and selecting by the central server a secret n×n matrix S as the secret master key, where S is selected as a small element following certain error distribution κ_(n) ₂ ; (2) selecting by the central server a random element M following uniform distribution, but making sure that M has an inverse: if the central server could not find one first time, it will try again till it finds one; then computing by the central server M ₁ =MS+te, where e is small following certain error distribution κ_(n) ₂ ; (3) then publicizing by the central server M and M₁ as the master public key; (4) assigning by the central server for each user indexed by i an public ID as A_(i), where A_(i) is small following certain error distribution κ_(n) ₂ , and the central server has a choice to generate it information that can identify the user i; (5) giving by the central server for each user, namely, the User i, a secret key: S _(i) =SA _(i) +tM ⁻¹ e _(i), where e_(i)'s entries are small following the error distribution κ; (6) then establishing by anyone using the ID, A_(i), and the master public key, a new public key for the user with ID A_(i), which is given as the pair (A_(i), B_(i)), where A_(i)=M and B _(i) =M ₁ A _(i) =MSA _(i) +teA _(i); and using by anyone as the public key to encrypt any message use the MLWE encryption system described in the Description section.
 4. The method according to claim 1, wherein q is a polynomial function of degree 2 or higher, or a similar function, and κ_(n) ² to is the a distribution that each component are independent and each component follow certain error distribution like the discrete error distribution κ_(σ), namely a discrete normal distribution over F_(q) center around 0 with standard deviation approximately √{square root over (n)}, or a similar distribution.
 5. The method according to claim 2, wherein q is a polynomial function of degree 2 or higher, or a similar function, κ_(n) ² is the a distribution that each component are independent and each component follow certain error distribution like the discrete error distribution κ_(σ), namely a discrete normal distribution over F_(q) center around 0 with standard deviation approximately √{square root over (n)} or a similar distribution.
 6. The method according to claim 3, wherein q is a polynomial function of degree 2 or higher, or a similar function, κ_(n) ² is the a distribution that each component are independent and each component follow certain error distribution like the discrete error distribution κ_(σ), namely a discrete normal distribution over F_(q) center around 0 with standard deviation approximately √{square root over (n)} or a similar distribution.
 7. The method according to claim 1, wherein the matrices is rectangular as long as the matrix multiplication is compatible and the parameters are adjusted accordingly.
 8. The method according to claim 2, wherein the matrices is rectangular as long as the matrix multiplication is compatible and the parameters are adjusted accordingly.
 9. The method according to claim 1, wherein the matrices are replaced with elements of the ring R_(q)=F_(q)[x]/f(x) with f(x)=x^(n)+1 and the parameters is adjusted accordingly.
 10. The method according to claim 2, wherein the matrices are replaced with elements of the ring R_(q)=F_(q)[x]/f(x) with f(x)=x^(n)+1 and the parameters is adjusted accordingly.
 11. The method according to claim 3, wherein the matrices are replaced with elements of the ring R_(q)=F_(q)[x]/f(x) with f(x)=x^(n)+1 and the parameters is adjusted accordingly.
 12. The method according to claim 2, wherein the procedure for two users i and j to derive a shared key is modified such that the roles of i and j are exchanged.
 13. The method according to claim 2, wherein several central servers to work together to build a distributed KD system.
 14. The method according to claim 3, wherein several central servers to work together to build a distributed IBE system.
 15. The method according to claim 3, wherein the procedure is extended further to build a hierarchical IBE system, where each user servers as a lower level central server.
 16. The method according to claim 1, wherein the rounding technique is replaced with a similar technique.
 17. The method according to claim 1, wherein the matrices are replaced with elements of the ring R_(q)=F_(q)[x]/f(x) with f(x)=x^(n)+1, the parameters is adjusted accordingly, and the polynomial elements used are selected in the form of f(x)=Πf_(i)(x)+g(x), where each f_(i), g(x) is a sparse matrix with very few terms terms none-zero.
 18. m
 18. The method according to claim 2, wherein the matrices are replaced with elements of the ring R_(q)=F_(q)[x]/f(x) with f(x)=x^(n)+1, the parameters is adjusted accordingly, and the polynomial elements used are selected in the form of f(x)=Πf_(i)(x)+g(x), where each f_(i), g(x) is a sparse matrix with very few terms terms none-zero.
 19. m
 19. The method according to claim 3, wherein the matrices are replaced with elements of the ring R_(q)=F_(q)[x]/f(x) with f(x)=x^(n)+1, the parameters is adjusted accordingly, and the polynomial elements used are selected in the form of f(x)=Πf_(i)(x)+g(x), where each f_(i), g(x) is a sparse matrix with very few terms terms none-zero. 